Conservation Properties of the Trapezoidal Rule in Linear Time Domain Analysis of Acoustics and Structures

Abstract

The trapezoidal rule, which is a special case of the Newmark family of algorithms, is one of the most widely used methods for transient hyperbolic problems. In this work, we show that this rule conserves linear and angular momenta and energy in the case of undamped linear elastodynamics problems, and an “energylike measureâ€‌ in the case of undamped acoustic problems. These conservation properties, thus, provide a rational basis for using this algorithm. In linear elastodynamics problems, variants of the trapezoidal rule that incorporate “highfrequencyâ€‌ dissipation are often used, since the higher frequencies, which are not approximated properly by the standard displacementbased approach, often result in unphysical behavior. Instead of modifying the trapezoidal algorithm, we propose using a hybrid finite element framework for constructing the stiffness matrix. Hybrid finite elements, which are based on a twofield variational formulation involving displacement and stresses, are known to approximate the eigenvalues much more accurately than the standard displacementbased approach, thereby either bypassing or reducing the need for highfrequency dissipation. We show this by means of several examples, where we compare the numerical solutions obtained using the displacementbased and hybrid approaches against analytical solutions.